If a = 3 square root of 3 in the right triangle below, what is the value of b?

The twο pοssible values οf ∠R tο the nearest 10th οf a degree are 33.6° and 326.4°.

A triangle is a clοsed twο-dimensiοnal geοmetric shape with three straight sides and three angles. It is the simplest pοlygοn, which is a flat shape cοnsisting οf straight lines.

We can use the Law οf Cοsines tο find the length οf side QR:

[tex]c^2 = a^2 + b^2 - 2ab[/tex] cοs(C)

where c is the length οf side QR, a is the length οf side PQ (which is unknοwn), b is the length οf side PR (which is 7.8 cm), and C is the angle οppοsite side c (which is 30°). Substituting the given values, we get:

[tex]QR^2 = PQ^2 + 7.8^2 - 2(PQ)(7.8)cos(30^\circ )[/tex]

[tex]QR^2 = PQ^2 + 60.84 - 7.8PQ[/tex]

Next, we can use the Law οf Sines tο relate the length οf side PQ tο the angle οppοsite it, ∠P:

PQ/sin(30°) = QR/sin(P)

PQ = QR(sin(30°)/sin(P))

Substituting this expressiοn fοr PQ intο the equatiοn fοr [tex]QR^2[/tex] abοve, we get:

[tex]QR^2 = [QR(sin(30^\circ)/sin(P))]^2 + 60.84 - 7.8[QR(sin(30^\circ)/sin(P))][/tex]

Simplifying and rearranging, we get a quadratic equatiοn in terms οf QR:

[tex]QR^2 - 3.9QR + 28.99 = 0[/tex]

Using the quadratic fοrmula, we find that:

QR ≈ 7.466 cm οr QR ≈ 3.866 cm

Since we knοw that QR < PQ + PR = 6 + 7.8 = 13.8, the οnly valid sοlutiοn is QR ≈ 7.466 cm. Therefοre, we have:

[tex]cos(R) = (PQ^2 + QR^2 - PR^2)/(2PQQR)[/tex]

[tex]cos(R) = (PQ^2 + 7.466^2 - 7.8^2)/(2PQ(7.466))[/tex]

[tex]cos(R) = (PQ^2 - 4.928)/[2PQ(7.466)][/tex]

Since cοs(R) ≤ 1, we have:

[tex]PQ^2 - 4.928 ≤ 2PQ(7.466)[/tex]

Sοlving fοr PQ using the quadratic fοrmula, we get:

PQ ≤ 5.474 cm οr PQ ≥ 20.032 cm

Since PQ < PR, the οnly valid sοlutiοn is:

PQ ≈ 5.474 cm

Nοw we can use the Law οf Cοsines again tο find ∠R:

[tex]cos(R) = (PQ^2 + PR^2 - QR^2)/(2PQPR)[/tex]

[tex]cos(R) = (5.474^2 + 7.8^2 - 7.466^2)/(2(5.474)(7.8))[/tex]

cοs(R) ≈ 0.828

R ≈ 33.6° οr R ≈ 326.4°

Therefοre, the twο pοssible values οf ∠R tο the nearest 10th οf a degree are 33.6° and 326.4°.

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The radioactive decay of sm-151 (an isotope of samarium) can be modeled by the differential equation dy /dt = -0.0077y, DOthe half-life of Sm-151 is 90 years.

      Sm-151 is a radioactive isotope of the element Samarium. The symbol for Sm-151 is 151Sm, and the atomic number of Samarium is 62. This isotope has a half-life of 88 years.

     Differential equations differential equation that model the radioactive decay of Sm-151 is given as

          dy/dt = -0.0077y, where t is measured in years.

To find the half-life of Sm-151, we can use the formula for half-life, which is given as:

                 t1/2 = (ln 2) / k

       Where k is the decay constant. To find k, we can use the given differential equation.

              dy/dt = -0.0077y

Separating variables, we get

            dy / y = -0.0077 dt

Integrating both sides,

        we get ln y = -0.0077 t + C

      Where C is the constant of integration.

To find C, we use the initial condition, y(0) = y0, where y0 is the initial amount of Sm-151.

Substituting this in the above equation, we get ln  y0 = CSo,

        the equation becomes y = -0.0077 t + ln y0

Taking the exponential of both sides, we get y = y0 e^(-0.0077t)

      Using the formula for k, we get k = 0.0077

      Substituting this in the formula for half-life,

          we get: t1/2 = (ln 2) / k

                              = (ln 2) / 0.0077

                              = 90 years

  Therefore, the half-life of Sm-151 is 90 years.

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The mean value and the variance of z are 3.5 and 0.486, respectively.

Suppose that a die is rolled twice and the average of the two numbers of spots is recorded as a quantity z. What are the mean value and the variance of z?

Let X be the first number of spots and Y be the second number of spots. As X and Y are independent and have the same distribution, we have

E(X) = E(Y) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5,
Var(X) = Var(Y) = (1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2) / 6 - (21 / 6)^2 = 2.9167.

Let Z = (X + Y) / 2 be the quantity of interest. Then

E(Z) = E[(X + Y) / 2] = E(X) / 2 + E(Y) / 2 = 3.5,
Var(Z) = Var[(X + Y) / 2] = Var(X) / 4 + Var(Y) / 4 = 0.486.

Therefore, the mean value and the variance of z are 3.5 and 0.486, respectively.

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The most important thing to notice about a table of q-sort correlates is d. general patterns that emerge.

Q-sorting is a method of ranking items in order of preference or importance, and Q-sort correlates are used to identify relationships between the ranked items. The correlations in the table represent how strongly the items are associated with each other based on their rankings.

While the exact value of the correlations is important for statistical analysis, the most valuable information in a table of q-sort correlates is the general patterns that emerge. These patterns can help to identify groups of items that are closely related to each other, as well as those that are more distantly related.

Understanding the wording of specific items may also be important, as it can impact how the items are ranked and how they relate to each other. However, the primary focus of a table of q-sort correlates is on the relationships between the items, rather than the items themselves.

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The value of the missing angles of the quadrilateral are:

a = 110°

b = 70°

c = 20°

Theorem sum of angles in a triangle states that they sum up to 180 degrees. Thus:

d = 180 - (78 + 32)

d = 70°

Similarly:

e = 180 - (78 + 32 + 18)

e = 20°

Sum of angles on a straight line is 180 degrees. Thus:

a = 180 - 70

a = 110°

We can see that e + b will be equal to 90 degrees because of the definition of right angles. Thus:

b = 90° - 20°

b = 70°

From sum of angles in a triangle as 180° is:

c = 180 - (90 + 70)

c = 20°

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Step-by-step explanation:

The circumference of a circle is given by the formula:

C = πd

where d is the diameter of the circle and π is a mathematical constant approximately equal to 3.14159.

In this case, the diameter of the circle is 58, so we can substitute this value into the formula:

C = πd

C = π(58)

C ≈ 182.09

Therefore, the circumference of the circle with diameter 58 is approximately 182.09 units. Note that the units of measurement were not specified in the question, so the units of the answer will depend on the units used for the diameter.

The skewness coefficient is a measure of asymmetry in a distribution that can be used to compare means and compare one sample to a known reference distribution, but it is not typically used to compare standard deviations or compare two samples with different measurement units.

The skewness coefficient is a measure of the degree of asymmetry in a probability distribution. It indicates the direction and degree of skewness in a distribution, and can be used to compare means and compare one sample to a known reference distribution.

A positive skewness coefficient indicates that the distribution has a longer tail on the right side and that the mean is greater than the median, while a negative skewness coefficient indicates that the distribution has a longer tail on the left side and that the mean is less than the median.

The skewness coefficient can be useful in detecting departures from normality and in identifying outliers that might affect statistical inference.

However, the skewness coefficient is not typically used to compare standard deviations or to compare two samples with different measurement units.

Comparing standard deviations involves analyzing the variability in the data, whereas comparing means and comparing one sample to a known reference distribution involves analyzing the central tendency of the data.

Additionally, comparing two samples with different measurement units requires converting the measurements to a common scale, which is not related to skewness.

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The total length of Joe's shape will be the sum of these lengths, which is 12 + 9 + 9 = 30cm.

To find the total length of Joe's shape, we first need to determine the length of each part of the strips.

For the first strip, which is divided into three equal parts, each part will be 36/3 = 12cm long. For the second strip, which is divided into four equal parts, each part will be 36/4 = 9cm long.

To make the shape, Joe will need to use two strips. Let's assume that Joe places the strip with three equal parts horizontally and the strip with four equal parts vertically. He will then need to cut each strip into the appropriate lengths to create the shape.

To create the shape, Joe will need to use one 12cm length from the first strip and two 9cm lengths from the second strip. He will then need to connect these lengths together to create the shape.

These lengths will add up to a total length for Joe's shape of 12 + 9 + 9 = 30 cm.

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Answer:

The first step is to calculate the interest that accrues over the 3-month period:

Interest = Principal x Rate x Time

= $4,000 x 0.075 x (3/12)

= $75

The amount due on the loan is the sum of the principal and interest:

Amount due = Principal + Interest

= $4,000 + $75

= $4,075

Rounding to the nearest cent gives: $4,075.00

There is a probability of around 60% that during a week picked randomly, the combined sum of their earnings will be below $0.2533.

The given probability is a measure of the chances that the occurrence of an event will happen. In the context of the question, we want to calculate the combined total of earnings that are less than a certain amount. We know that the probability is approximately 0.6. Therefore, the answer is given by the value that satisfies this condition.

That is, P(X < a) = 0.6where X represents the combined total earnings of a randomly selected week, and a is the required amount.

To solve for a, we can use the cumulative distribution function (CDF) of X. The CDF gives the probability that X is less than or equal to a. Therefore, CDF(a) = P(X ≤ a)

The complement of this probability isP(X > a) = 1 - CDF(a)

We know that P(X < a) = 0.6, which is equivalent to (X ≤ a) = P(X < a) + P(X = a) = 0.6 + 0 = 0.6

Therefore, P(X > a) = 1 - CDF(a) = 1 - P(X ≤ a) = 1 - 0.6 = 0.4

Now we can use a calculator to find the value of a that corresponds to P(X > a) = 0.4.

For example, if we use a calculator, we get a value of a = 0.2533 (rounded to four decimal places).

Therefore, the probability is approximately 0.6 that in a randomly selected week, they will make a combined total of less than $0.2533.


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The number of square units enclosed by the polygon is 18.

First, count the number of dots that are enclosed by the polygon (including those on the boundary). There are 14 such dots.Next, count the number of dots on the boundary of the polygon. There are 8 dots on the boundary.Each of the dots on the boundary corresponds to a line segment of the polygon.

So, the perimeter of the polygon is 8 units (the length of each of these line segments).Now, we can use Pick's theorem to find the area of the polygon. Pick's theorem states that A = i + b/2 - 1, where A is the area of the polygon, i is the number of dots inside the polygon, and b is the number of dots on the boundary of the polygon.

So, plugging in the values we have: A = 14 + (8/2) - 1 = 14 + 4 - 1 = 17

Therefore, the area of the polygon is 17 square units.However, we have to remember that the dots are spaced one unit apart, horizontally and vertically.

Therefore, each square that is enclosed by the polygon has an area of 1 square unit. We counted 17 such squares, so the total area enclosed by the polygon is 17 square units.

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Answer:

⅔h – 5 = 11

Step-by-step explanation:

         h = babysitting hours last week

     ⅔h = two-thirds of those hours

⅔h - 5 = five hours fewer than two-thirds of those hours

⅔h – 5 = 11

The equation is ⅔h – 5 = 11.

As the given abc is a right triangle with points a(2, 5), b(2, 0), and c(x, 0). The area of 'abc' is 25 square units. and the equation of AC is x = 5√5.

The equation of ac has to be found. Since triangle ABC is right-angled, it can be shown that

                  AC² = AB² + BC²

    Finding the lengths of AB and BC before using the Pythagorean theorem to find AC.

    Using the distance formula,

                     AB =√(5²+0²)

                           = √25

                          = 5,

           And BC = √(x²+0²)

                         = √x²

                          = x.

  Area of a right triangle = ½(base × height)

          Here base = AB and height = BC,

              So,  ½ × 5 × x = 25

                            ⇒ 5x = 50

                              ⇒ x = 10

       Now, to find AC, use the Pythagorean Theorem.

                      AC² = AB² + BC²

                             = 5² + 10²

                             = 25 + 100

                             = 125AC

                             = √125

                             = 5√5

     Thus, the equation of AC is x = 5√5. Answer: x = 5√5.

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We can solve the given system of equations to find how many grams of fat are in each when two eggs and two pieces of bacon have a total of 22 grams of fat and two eggs and four pieces of bacon have a total of 32 grams of fat.

Step-by-step explanation:

Let's consider that each piece of bacon has "x" grams of fat and each egg has "y" grams of fat.

From the given information, we can form two equations as follows:

Equation 1: 2x + 2y = 22 (when two eggs and two pieces of bacon have a total of 22 grams of fat)

Equation 2: 4x + 2y = 32  (when two eggs and four pieces of bacon have a total of 32 grams of fat)

To solve for "x" and "y", let's perform elimination by multiplying Equation 1 by (-2) and adding it to Equation 2.

-4x - 4y = -44  (multiply Equation 1 by -2)
+ 4x + 2y = 32  (Equation 2)

-2y = -12

y = 6

Substitute the value of "y" in Equation 1 or 2 to solve for "x":

2x + 2y = 22

2x + 2(6) = 22

2x = 10

x = 5

Therefore, each piece of bacon has 5 grams of fat and each egg has 6 grams of fat.

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The total amount paid by Jennifer for 14.75 gallons of gasoline at the cost of $2.95per gallons is equal to $43.51.

Total gallons of gasoline bought by Jennifer for her car = 14.75 gallons

Cost of gasoline per gallons = $2.95

The total amount Jennifer paid for the gasoline is equal to,

Amount paid by Jennifer

= Number of gallons of gasoline x Cost per gallon

Substitute the value in the formula we get,

⇒ Amount paid by Jennifer = 14.75 gallons x $2.95/gallon

⇒ Amount paid by Jennifer = $43.5125

⇒ Amount paid by Jennifer = $43.51

Therefore, the closest amount Jennifer paid for the gallons of gasoline is equal to $43.51.

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When the tank is tilted, the water level reaches the midpoint of the 60 cm side, which is 30 cm from the bottom of the tank. This means that the height of the water is 30 cm.

To find the depth of the water when the tank is returned to its horizontal position, we need to use the concept of similar triangles. The two triangles formed by the water level and the sides of the tank are similar, because they have the same angles.

Let x be the depth of the water in centimeters when the tank is returned to its horizontal position. Then we have:

(40 - x) / 60 = 30 / 50

Simplifying this equation, we get:

40 - x = 36

x = 4

Therefore, the depth of the water in the fish tank when it is returned to its horizontal position is 4 cm.

The probability that Emme answers exactly 3 questions correctly is 0.1961.

Since each question has four answer choices and only one is correct, the probability of Emme randomly guessing the correct answer for each question is 1/4 = 0.25. Emme needs to answer exactly 3 out of the 9 questions correctly, so we use the binomial distribution with n = 9 and p = 0.25 to calculate the probability:

P(X = 3) = (9 choose 3) * (0.25)^3 * (0.75)^6

Using a calculator, we get:

P(X = 3) = 0.1961

Therefore, the probability that Emme answers exactly 3 questions correctly is 0.1961.

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Asking questions in math class is crucial for deepening understanding, and teachers strive to create a supportive learning environment for students to feel comfortable seeking help.

Students may hesitate to ask questions in math class for various reasons, such as fear of being judged by peers or the teacher, feeling like their question is not important, or simply not wanting to disrupt the flow of the lesson. However, it's essential to remember that asking questions is an integral part of the learning process and can lead to a deeper understanding of the subject.

When students do ask questions, they may feel relieved to have their doubts clarified, and it can also benefit other students who may have had the same question but were hesitant to ask. Teachers typically encourage questions and strive to create a supportive learning environment where students feel comfortable asking for help.

In summary, asking questions is an essential aspect of learning, and students should feel encouraged to ask them, as it can lead to a deeper understanding of the subject and help their peers who may have had similar questions.

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From the given information provided, the expression that need to determine the total amount is Total cost = $2.85/gallon x g gallons + $3.15/can x c cans.

The expression that can be used to determine the total amount Joe spent on gasoline and oil is:

Total cost = Cost of gasoline + Cost of oil

We can represent the cost of gasoline as:

Cost of gasoline = price per gallon x number of gallons

Substituting the given values, we get:

Cost of gasoline = $2.85/gallon x g gallons

Similarly, we can represent the cost of oil as:

Cost of oil = price per can x number of cans

Substituting the given values, we get:

Cost of oil = $3.15/can x c cans

Putting it all together, we get:

Total cost = $2.85/gallon x g gallons + $3.15/can x c cans

Expression that can be used to determine the total amount Joe spent on gasoline and oil is:

Total cost = $2.85/gallon x g gallons + $3.15/can x c cans

Question - Joe bought g  gallons of gasoline for $2.85 per gallon and c  cans of oil for  $3.15 per can. What expression can be used to determine the total amount Joe spent on gasoline and oil?

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If each pencil cost $1.50 and Alicia had $22 to spend on them, she could purchase 14 pencils. This situation's inequality is as follows: 1.5x ≤ 22.

To find out how many pencils Alicia can buy, we can divide her total budget by the cost per pencil:

Number of pencils = Total budget / Cost per pencil

Number of pencils = $22 / $1.50

Number of pencils = 14.67 (rounded to two decimal places)

Alicia cannot buy a fraction of a pencil, so she can buy a maximum of 14 pencils.

The inequality that represents this situation is:

1.5x ≤ 22

Where x represents the number of pencils Alicia can buy. We divide both sides of the inequality by 1.5 to isolate x:

x ≤ 14.67

Since Alicia cannot buy a fraction of a pencil, we round down to the nearest whole number and get:

x ≤ 14

So, Alicia can buy a maximum of 14 pencils.

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We can find angle z by using triangle interior theorem

All angles in a triangles add up to 180 so

[tex]z + 82 +90 = 180[/tex]

[tex]z = 8[/tex]

To find c, we know c is the hypotenuse and we are given a base of 5, we can use cosine

[tex] \cos(82) = \frac{5}{c} [/tex]

[tex]c = \frac{5}{ \cos(82) } [/tex]

Use a calculator, and we get

[tex]c = 35.92[/tex]

To find a, we can use tangent

[tex] \tan(82) = \frac{a}{5} [/tex]

[tex]5 \tan(82) = a[/tex]

[tex]a = 35.58[/tex]

There is a 2.28% likelihood that the average weight of the bags in a randomly selected pack is 2 or more grams lighter than advertised.

Assume that the advertised weight is equivalent to the typical weight of one bag, which is 30 grammes. A pack of nine bags would therefore weigh 9 x 30 = 270 grammes.

Let's now assume that each bag actually weighs 30 grammes on average, with a 2 gramme standard deviation, according to a normal distribution (which represents the variability in the weight of the bags).

We need to apply the central limit theorem to determine the probability that the average weight of the bags in a randomly selected pack is 2 or more grammes less than stated. According to this theorem, the mean of a sufficiently large sample (in this case, nine bags) will have a sampling distribution that is roughly normal, with a mean of 30 grammes and a standard deviation of 2 grammes divided by the square root of the sample size (9 = 0.67 grammes). The mean will also be equal to the population's standard deviation.

We can determine the likelihood that the sample mean of a pack of 9 bags weighs less than 28 grammes using a normal distribution table or calculator (which is 2 grammes less than the advertised weight of 30 grams). This likelihood is roughly 0.0228, or 2.28%.

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qwerty

2.8% the answer is 2.8% and b is 2.8%

Two events that are collectively exhaustive are event 1: a value of 6 or more and event 2: a value of 7 or less.

These events are collectively exhaustive because the events cover the entire possible range of values of the dice rolls. If one of these events does not occur, then the other event must occur. This means that the sum of the dice rolls is either 6 or less, or it is greater than 6. This means that the sum of the dice rolls is either 7 or more, or it is less than 7.

Another set of events that are collectively exhaustive are event 1: rolling an even number and event 2: rolling an odd number. These events are collectively exhaustive because the events cover the entire set of possible outcomes of rolling two dice. If one of these events does not occur, then the other event must occur. This means that the sum of the dice rolls is either even or odd. This means that the sum of the dice rolls is not both even and odd at the same time.

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Method 1: Graphing

Graphing is best when the both equations are shown in slope-intercept form. For example:

[tex]y=\frac{1}{2} x+2\\y=-\frac{1}{2} x+4[/tex]

[tex](2,3)[/tex]

You can easily graph this system of equations and find the intersecting point. The graph is displayed below. This may not always be the case, however and you may get complicated fractions in your answer. In this case, substitution or elimination may be better.

Method 2: Substitution

Substitution is a good method to solve a system of equations when one of the equations can be rearranged to isolate one variable, or the equation already solves for a variable. This isolated expression can then be substituted into the other equation(s) to create a new equation(s) with only one variable.

For example, consider the system of equations:

[tex]y=x-2\\2x-5y=1[/tex]

Since y is much easier to substitute, we can choose y to substitute into the other equation:

[tex]2x-5(x-2)=1[/tex]

We can then simplify the equation:

[tex]2x-5x+10=1\\-3x=-9\\x=3[/tex]

Then you can substitute x into the original equation:

[tex]y=3-2\\y=1[/tex]

Solution: [tex](3,1)[/tex]

Substitution can be a useful method when the system involves two or three variables and one equation can be easily rearranged to isolate a variable. However, it can become more difficult or time-consuming when the system involves more variables or when the equations are not solving for a variable or can be easily solved. In these cases, other methods such as elimination is more efficient.

Method 3: Elimination

Elimination is a good method to solve a system of equations when the equations can be added or subtracted in a way that eliminates one of the variables.

For example, consider the system of equations:

[tex]x+y=90\\x-y=18[/tex]

We can cancel out the variable y and add the other numerals and variables:

[tex]2x=108[/tex]

This simplifies to:

[tex]x=54[/tex]

We can then solve for y:

[tex]54+y=90\\y=36[/tex]

Solution: [tex](54, 36)[/tex]

Once we have the value of x, we can substitute it back into one of the original equations to solve for y.

Although elimination is slightly more complicated, it is the most efficient method out of all of the three methods I have shown here.

Hope this helped you :)

(This took like 30 minutes ;-;)

The size of angle DAB in the quadrilateral is 49°.

The sum of the interior angles of a quadrilateral is 360°. We can say:

∠A + ∠B + ∠C + ∠D = 360°

∠A + 34° + ∠C + 228° = 360°

∠A  + ∠C + 262° = 360°

∠A  + ∠C = 360 - 262

∠A  + ∠C = 98

Since angles DAB and BCD are the same size. This implies ∠A  = ∠C. Thus:

∠A  + ∠A = 98

2∠A  = 98

∠A = 98/2

∠A  = 49°

Therefore, the size of angle DAB is 49°.

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Complete Question

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The total amount of money earned by Hudson in a day on which he waits on 18 tables is $264.

The total amount of money earned by Hudson in a day on which he waits on t tables is $(30 + 13t).

Based on the information provided above, we can logically deduce that a linear equation that models the total amount of money earned by Hudson at this restaurant is represented by the following function:

y = 30 + 13t

Where:

When the number of tables he waits on that day is equal to 18, Hudson's total earnings can be calculated as follows;

y = 30 + 13t

y = 30 + 13(18)

y = $264.

When the number of tables he waits on that day is t tables, Hudson's total earnings can be calculated as follows;

y = 30 + 13t = $(30 + 13t).

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The inequality that represents Miguel's spending limit is $16.00 ≤ x.

Let x represent the total amount of money Miguel wants to spend. We know he spent $10.49 on popcorn and can spend up to $5.51 on a drink.

To find the inequality, we can add these two amounts together and set it less than or equal to x, since x represents the maximum amount he wants to spend. Mathematically, we can write:

$10.49 + $5.51 ≤ x

Simplifying this inequality, we get:

$16.00 ≤ x

This means that Miguel can spend up to $16.00 in total on the popcorn and drink combined. If he spends more than $16.00, he will have exceeded his limit.

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Answer:

12π feet

Step-by-step explanation:

The formula for the area of a circle is A = πr², where A is the area and r is the radius. We are given that the area is 36π ft², so we can set up an equation:

36π = πr²

To solve for the radius, we can divide both sides by π:

36 = r²

Taking the square root of both sides, we get:

r = 6

Now that we know the radius is 6 feet, we can use the formula for the circumference of a circle, C = 2πr:

C = 2π(6)

Simplifying, we get:

C = 12π

Therefore, the circumference of the circle is 12π feet.